Acquisition methods, in which multiple seismic vibratory sources, commonly called vibrators, are operated at the same time and the separate seismic seismograms then recovered, can considerably reduce the time it takes to record vibroseis data thereby reducing the cost of acquiring seismic surveys. Similarly, computer modeling of multiple source data followed by recovery of the separate seismograms from the modeled data is a computationally efficient alternative to simulating separate seismograms. However, the seismograms obtained from such acquisition methods are contaminated by more noise than seismograms obtained from surveys in which single vibrators are used alone, or from surveys in which multiple vibrators and multiple sweeps, with the number of sweeps greater than or equal to the number of vibrators, are used. The additional noise, termed interference noise, arises because multiple seismograms need to be recovered or separated from a smaller number of vibroseis field records, so that a separate seismogram for each individual vibrator is obtained. As may be expected, the computer-simulated seismograms also suffer from the aforementioned interference noise problem.
Vibrator Field Records and Seismograms
Seismic vibrators have long been used in the seismic data acquisition industry to generate the acoustic signals needed in geophysical exploration. The conventional use of vibrators involves several well-understood steps. First, one or more vibrators are located at a source point on the surface of the earth or one or more marine vibrators are deployed in water. Second, the vibrators are activated for several seconds, typically ranging from four to thirty-two, with a pilot signal. The pilot signal is typically a sweep function that varies in frequency during the period of time in which the vibrators are activated. Third, seismic receivers are used to receive and record response data for a period of time equal to the sweep time plus a listen time. Typically, the listen time is greater than the time needed for the seismic wave to travel to the deepest target of interest, reflect from the target of interest and travel to the detectors. The response data measured and recorded vs. time at each receiver is called a trace and the group of traces is called a record. The period of time over which data are recorded includes at a minimum the sweep plus the time necessary for the seismic signals to travel to and reflect off of the target reflectors of interest and for the reflected signals to return to the receivers. Fourth, seismograms, similar to those recorded with impulsive sources, are conventionally generated by cross correlating the recorded data with either the pilot signal or a reference sweep. Fifth, the sweep and correlation steps are repeated several times, typically four to eight, and the correlations are added together in a process referred to as stacking, the purpose of which is to enhance signals and reduce noise. Finally, the vibrators are moved to a new source point and the entire process is repeated.
A useful construct in considering this application of seismic vibrators is the standard convolutional model (Sheriff, Encyclopedic Dictionary of Applied Geophysics, 4th Ed., 67 (2002)). This model represents each seismic trace as a convolution of the earth reflectivity function or earth impulse response with a seismic wavelet. The process of convolution is equivalent to the process of applying a filter and is designated with the symbol  . In the case of a seismic vibrator, a field data trace d(t) can be represented by the convolution of the earth response e(t) with the sweep function s(t), which is several seconds long, plus noise n(t),d(t)=s(t)e(t)+n(t)  (1)Thus, field records can be considered as a superposition of multiple reflections of the sweep function with the reflections occurring at different times. In the frequency domain, Equation 1 becomesD(ƒ)=S(ƒ)E(ƒ)+N(ƒ)  (2)In Equation 2, capital lettered symbols such as S(ƒ) denote the Fourier transforms of the respective small lettered symbols such as s(t).
Other types of geophysical surveys such as electromagnetic surveys and multi-component seismic surveys are also governed by Equations (1) and (2). For example, in electromagnetic surveys, the s(t) in Equation (1) can be a sweep for a dipole source that radiates electromagnetic energy, e(t) would be the earth's diffusive response to the current dipole source, and the measurement d(t) would be an electric field. In multi-component surveys, the s(t) in Equation (1) can be a sweep for a seismic source, e(t) would be the earth's response to the source, and the measurement d(t) would be the particle motion measured along the horizontal and/or vertical direction. A multi-component survey typically means a seismic survey using multiple types of sensors in which each type of sensor measures a different direction of the vector particle to motion or the scalar pressure change.
Computer simulations of field records for all types of geophysical surveys are also exactly described by Equations (1) and (2). In computer modeled seismic data, the superposition of multiple reflections (see the s(t)e(t) term in Equation 2) is performed by running a modeling program that mimics the earth's response (for example, by solving a partial differential equation using finite difference method), and the noise term encapsulates the imperfections of the computer modeling. In contrast, during actual field acquisition, the superposition term and noise term in the field records are naturally generated by the earth in response to the vibratory source sweep signals. Further, in the computer modeled case, the sweep function is exactly known. In contrast, during field acquisition, the actual sweep function that probes the earth can differ from the designed pilot sweep (input to the vibratory sources), and hence needs to measured (termed measured ground force below) or inferred.
The invention is explained below mostly in terms of data acquired from actual seismic field operations where multiple, simultaneously operating, seismic vibrators were the source device. However, all concepts disclosed by such examples are equally applicable to all types of geophysical data and their computer-simulation, unless specified. For example, the term seismograms used routinely in the descriptions below would refer to band-limited Green's functions in computer modeling and to measured electromagnetic energy in electromagnetic surveys. In the electromagnetic case, a vibratory source refers to an electromagnetic source (such as a current dipole source) that is active for continuous periods of time.
Ideally, it would be desirable to extract the full bandwidth earth response E(ƒ) from the recorded data D(ƒ). However, this is not possible in practice because the sweeps used in seismic exploration are band-limited. Instead one seeks to extract the earth response that is convolved with known large bandwidth wavelet with much smaller time extent (compared to the time extent of the sweep function). Such an approximation to the earth response is called a seismogram.
In the described application of seismic vibrators, seismograms g(t) are generated by cross correlating the field record traces measured at various receiver locations with the pilot signal or reference sweep. This cross-correlation step computes the similarity between the field record traces and the sweep function and yields an approximation or estimate of the earth reflectivity function. The process of correlation can be written as a convolution filter, in which the filter is the time reverse of the sweep function as in Equation 3.g(t)=s(−t)d(t)=[s(−t)s(t)]e(t)+s(−t)n(t).  (3)In the frequency domain, Equation 3 can be equivalently expressed asG(ƒ)=S*(ƒ)D(ƒ)=[S*(ƒ)S(ƒ)]E(ƒ)+S*(ƒ)N(ƒ),  (4)where S* is the complex conjugate of S. The resulting seismogram G(ƒ) in Equation 4 is in the form of the convolutional model in which the earth reflectivity function is convolved with a wavelet given by the autocorrelation function [S*(ƒ)S(ƒ)] of the sweep function S(ƒ). The autocorrelation of the sweep function is essentially non-zero for only a few hundreds of milliseconds. Thus, it approximates the wavelet of an impulsive source better than the sweep wavelet, which is several seconds long.
Instead of using the pilot sweeps and correlation filters to obtain the seismograms as in Equation 4, U.S. Pat. No. 5,550,786 to Allen discloses the use of measured motion from accelerometers on the vibrators to derive an inverse filter. Typically, the so-called ground-force signal, which is a mass-weighted sum of measured accelerometer signals on the reaction mass and on the base plate, is used. The ground-force signal is conventionally used in feedback electronics on the vibrator and is a better approximation for the actual sweep signal imparted into the ground because it includes harmonics generated by nonlinearities in the vibrator mechanics and in the soil. Thus, it can be called a vibrator signature. Other sweep signals derived from measurements can also be used, and they can be used to correlate the field record traces or to construct a filter to remove the signature or sweep signals and replace them with a wavelet.
Wavelets other than the sweep autocorrelation wavelet can be used as well. Trantham (U.S. Pat. No. 5,400,299) and Krohn (International Publication No. WO 2004/095073 A2) apply deconvolution filters that remove or divide the vibrator signature S(ƒ) from the data and replace it with the desired wavelet W(ƒ). In the frequency domain, this operation can be expressed using Equations 5 and 6.
                                                                        G                ⁡                                  (                  f                  )                                            ⁡                              [                                                      W                    ⁡                                          (                      f                      )                                                                            S                    ⁡                                          (                      f                      )                                                                      ]                                      ⁢                          D              ⁡                              (                f                )                                              =                                                    [                                                      W                    ⁡                                          (                      f                      )                                                                            S                    ⁡                                          (                      f                      )                                                                      ]                            ⁢                              S                ⁡                                  (                  f                  )                                            ⁢                              E                ⁡                                  (                  f                  )                                                      +                                          [                                                      W                    ⁡                                          (                      f                      )                                                                            S                    ⁡                                          (                      f                      )                                                                      ]                            ⁢                              N                ⁡                                  (                  f                  )                                                                    ;                            (        5        )                                          G          ⁡                      (            f            )                          =                                            W              ⁡                              (                f                )                                      ⁢                          E              ⁡                              (                f                )                                              +                                    [                                                W                  ⁡                                      (                    f                    )                                                                    S                  ⁡                                      (                    f                    )                                                              ]                        ⁢                                          N                ⁡                                  (                  f                  )                                            .                                                          (        6        )            The desired wavelet can be of the form of the autocorrelation of the sweep or another wavelet with a bandwidth similar to the sweep function. Typically, the wavelet is chosen to be a few hundred milliseconds long, so that the resulting seismogram resembles data recorded with an impulsive source such as dynamite.Methods to Reduce Time and Cost of Vibroseis Acquisition
The cost of acquiring vibroseis surveys largely depends on the time it takes to record the survey, which is in turn determined by the time required to record data at each source station. The recording time at each source station depends on the number of sweeps, the sweep length, and the listen time. The listening time (Sheriff, page 211) is the time after the cessation of the vibrator sweep until the end of the record. This time must be long enough for the last vibration to travel from the source through the earth formation to the receiver. For example, if four 8-second sweeps are performed and each sweep has a 7-second listening time, then at least 60 seconds are required to record data at each station. If multiple stations are recorded simultaneously, or if a single sweep is used instead of multiple sweeps, or if the listening times for which the vibrators are idle between sweeps are reduced or eliminated, then less time would be needed for recording the survey. Consequently, the overall cost of the survey would be reduced. While such methods can be used to record data in less time, often the quality of the separated seismograms is worse than that obtained from data recorded with one station at a time with multiple sweeps.
During computer modeling of field records, a key metric is to minimize the total computational cost for computing all the separate seismograms. As expected, all factors identified above that minimize the cost of acquisition of vibroseis surveys also minimize the computational cost of computer modeling.
Separation of Multiple Vibrator Records with Number of Sweeps Greater than or Equal to Number of Vibrators
When vibrators are located at different stations and when multiple sweeps are recorded, with the number of sweeps greater than the number of vibrators, then individual separated seismograms can be recovered from the field records by the High Fidelity Vibratory Seismic Method (“HFVS”) as first described by Sallas, et al. (see, for example, U.S. Pat. No. 5,721,710). The method uses the vibrator signatures (either the pilot or ground-force signals) for to each vibrator and for each sweep and solves a set of linear equations to design an optimal filter that separates the earth response for each vibrator. As long as there is at least the same number of sweeps and records as the number of unknowns (earth responses for each vibrator), then the solution is well posed. A filter that optimally separates the data into the different seismograms can be found. Consequently, noises arising due to interference between the different seismograms being simultaneously acquired are small.
The HFVS method is more fully described in association with FIG. 1, which depicts typical land-based data acquisition system geometry. FIG. 1 shows four vibrators 18, 20, 22, and 24, mounted on vehicles 34, 36, 38, and 40. The four different signatures transmitted into the ground during sweep i may be called si1, si2, si3, si4. The sweeps for each vibrator are designed to differ, typically in the phase of the sweep. Each signature is convolved with a different earth reflectivity sequence e1, e2, e3, e4 which includes reflections 26 from the interface 28 between earth layers having different impedances (the product of the density of the medium and the velocity of propagation of acoustic waves in the medium). A trace di recorded at a geophone 30 is a sum of the signature-filtered earth reflectivities for each vibrator. Formulating this mathematically, data trace di recorded for sweep i is:
                                                        d              i                        ⁡                          (              t              )                                =                                                    ∑                                  j                  =                  1                                N                            ⁢                                                                    s                    ij                                    ⁡                                      (                    t                    )                                                  ⊗                                                      e                    j                                    ⁡                                      (                    t                    )                                                                        +                                          n                i                            ⁡                              (                t                )                                                    ,                            (        7        )            where sij(t) equals the sweep i from vibrator j, ej(t) equals the earth reflectivity seen by vibrator j and  denotes the convolution operator. Equation 7 is based on the convolutional model explained above. This model is a consequence of the concept that each reflected seismic wave causes its own effect at each geophone, independent of what other waves are affecting the geophone, and that the geophone response is simply the sum (linear superposition) of the effects of all the waves. If each sweep sij(t) is repeated j=1 to M times, then in matrix terms, Equation 7 can be equivalently be expressed as
                              [                                                                                          d                    1                                    ⁡                                      (                    t                    )                                                                                                                                            d                    2                                    ⁡                                      (                    t                    )                                                                                                      ⋮                                                                                                          d                    M                                    ⁡                                      (                    t                    )                                                                                ]                =                                            [                                                                                                                  s                        11                                            ⁡                                              (                        t                        )                                                                                                                                                s                        12                                            ⁡                                              (                        t                        )                                                                                                  …                                                                                                      s                                                  1                          ⁢                                                                                                          ⁢                          N                                                                    ⁡                                              (                        t                        )                                                                                                                                                                                s                        22                                            ⁡                                              (                        t                        )                                                                                                                                                s                        22                                            ⁡                                              (                        t                        )                                                                                                  …                                                                                                      s                        22                                            ⁡                                              (                        t                        )                                                                                                                                  ⋮                                                        ⋮                                                        …                                                        ⋮                                                                                                                                      s                                                  M                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                        t                        )                                                                                                                                                s                                                  M                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                        t                        )                                                                                                  …                                                                                                      s                                                  M                          ⁢                                                                                                          ⁢                          2                                                                    ⁡                                              (                        t                        )                                                                                                        ]                        ⊗                          [                                                                                                                  e                        1                                            ⁡                                              (                        t                        )                                                                                                                                                                                e                        2                                            ⁡                                              (                        t                        )                                                                                                                                  ⋮                                                                                                              e                                              N                        ⁡                                                  (                          t                          )                                                                                                                                ]                                +                                    [                                                                                                                  n                        1                                            ⁡                                              (                        t                        )                                                                                                                                                                                n                        2                                            ⁡                                              (                        t                        )                                                                                                                                  ⋮                                                                                                                                      n                        M                                            ⁡                                              (                        t                        )                                                                                                        ]                        .                                              (        8        )            Thus, in the HFVS method, N vibrators radiate M≧N sweeps into the earth, resulting in M recorded data records. After the field data are recorded, the HFVS method involves finding an operator, by solving a set of linear equations based on the known M×N vibrator signatures that finds the set of N earth reflectivities that best predict the recorded data. In computer modeling, as expected, when N computer-simulated vibratory sources radiate M≧N sweeps, the computed M data records obtained by running a modeling program are also described using Equation (8).
In the frequency domain, that is, after Fourier transformation, the set of equations represented by Equation 7 can be written as
                                          D            i                    ⁡                      (            f            )                          =                                            ∑                              j                =                1                            N                        ⁢                                                            S                  ij                                ⁡                                  (                  f                  )                                            ⁢                                                E                  j                                ⁡                                  (                  f                  )                                                              +                                    N              ⁡                              (                f                )                                      .                                              (        9        )            In matrix form, for M sweeps and N vibrators and ignoring the noise term, Equation 8 can be written as
                                          [                                                                                                      S                      11                                        ⁡                                          (                      f                      )                                                                                                                                  S                      12                                        ⁡                                          (                      f                      )                                                                                        …                                                                                            S                                              1                        ⁢                                                                                                  ⁢                        N                                                              ⁡                                          (                      f                      )                                                                                                                                                              S                      21                                        ⁡                                          (                      f                      )                                                                                                                                  S                      22                                        ⁡                                          (                      f                      )                                                                                        …                                                                                            S                                              2                        ⁢                                                                                                  ⁢                        N                                                              ⁡                                          (                      f                      )                                                                                                                                                              S                      31                                        ⁡                                          (                      f                      )                                                                                                                                  S                      32                                        ⁡                                          (                      f                      )                                                                                        …                                                                                            S                                              3                        ⁢                                                                                                  ⁢                        N                                                              ⁡                                          (                      f                      )                                                                                                                                                              S                      41                                        ⁡                                          (                      f                      )                                                                                                                                  S                      42                                        ⁡                                          (                      f                      )                                                                                        …                                                                                            S                                              4                        ⁢                                                                                                  ⁢                        N                                                              ⁡                                          (                      f                      )                                                                                                                    ⋮                                                  ⋮                                                  …                                                  ⋮                                                                                                                        S                                              M                        ⁢                                                                                                  ⁢                        1                                                              ⁡                                          (                      f                      )                                                                                                                                  S                                              M                        ⁢                                                                                                  ⁢                        2                                                              ⁡                                          (                      f                      )                                                                                        …                                                                                            S                      MN                                        ⁡                                          (                      f                      )                                                                                            ]                    ⁡                      [                                                                                                      E                      1                                        ⁡                                          (                      f                      )                                                                                                                                                              E                      2                                        ⁡                                          (                      f                      )                                                                                                                    ⋮                                                                                                                        E                      N                                        ⁡                                          (                      f                      )                                                                                            ]                          =                              [                                                                                                      D                      1                                        ⁡                                          (                      f                      )                                                                                                                                                              D                      2                                        ⁡                                          (                      f                      )                                                                                                                                                              D                      3                                        ⁡                                          (                      f                      )                                                                                                                                                              D                      4                                        ⁡                                          (                      f                      )                                                                                                                    ⋮                                                                                                                        D                      M                                        ⁡                                          (                      f                      )                                                                                            ]                    .                                    (        10        )            Using vector notations, with overhead→denoting vectors and bold type to denote matrices, Equation 10 can be written asS{right arrow over (E)}={right arrow over (D)}.  (11)When the number of sweeps is greater than or equal to the number of vibrators, the system of simultaneous equations above can be solved for {right arrow over (E)}. For example, if the number of sweeps is equal to the number of vibrators, then{right arrow over (E)}=F{right arrow over (D)}  (12)where F is the filter or operator which when applied to the data separates it into individual vibrator seismograms. The filter F is equal to the inverse of the matrix SF=(S)−1.  (13)In the presence of noise, the above equality holds only approximately because damped inversion is typically employed especially outside the bandwidth of the sweep. Krohn (PCT Publication No. WO 2004/095073 A2) includes a method to separate the data and shape the data to a desired band-limited wavelet. In the preferred method, which uses a least squares solution, the set of seismograms are{right arrow over (G)}=WS*(S*S)−1{right arrow over (D)},  (14)where W is the desired wavelet.
The HFVS method is more fully described in FIGS. 2A-D, which show raw field records for HFVS acquisition. This example illustrates separation with four vibrators and four sweeps. During the first sweep, a group (41) of four vibrator signatures (the disturbances imparted into the ground by each of the four vibrators) are measured by accelerometers on the vibrators. These can differ from the pilot sweep by harmonics caused by the vibrator's mechanical components and their coupling to the ground. Alternatively, signatures may be determined by other near-source measurement, or be estimated, or the vibrator pilot sweep signature may be used in lieu of measurements. In addition, data traces, 45, from a number of geophones are recorded. After the 8-second sweep, the vibrator motion ceases. The record must continue after the cessation of the vibrator motion, 50, so that the last vibration can travel from the source through the earth formation to the receiver. This time is referred to as the listening time. The recordings are repeated three more times so there are four recordings of vibrator signatures (41, 42, 43, 44) and geophone data (45, 46, 47, 48). The sixteen vibrator signatures are used to design a filter that separates the data into four separated seismograms, one for each vibrator location. The separated seismograms for model data are shown in FIGS. 3A-D. Since there are as many data records as there are vibrator locations, the recovered seismograms, 51, 52, 53, 54, are well-separated and there is no interference noise.
Sometimes, multiple fleets of vibrators, with each fleet comprising several vibrators, can be used and separated with the HFVS separation method. In such a case, each fleet would be centered at a different station, and all the vibrators in the same fleet would be operated with identical sweeps. The fleets would be operated simultaneously with a different sweep pattern for each fleet. If there are as many sweeps as there are fleets, then the HFVS separation method can be used to obtain the individual seismograms for each fleet. However, it would not be possible to separate seismograms for the individual vibrators in a fleet because they have the same sweep as the other vibrators in the fleet. The fleet of vibrators would form a source array and there would be one seismogram per fleet at each station.
Separation of Multiple Vibrator Records with Number of Sweeps Fewer than the Number of Vibrators
The problem with the HFVS approach is that multiple sweeps are needed with each sweep followed by a listening time when the vibrators are idle. If fewer sweeps or, in the preferred case, one continuous sweep per vibrator can be used, then the data can be recorded in less time. Formulating the acquisition mathematically for a single continuous sweep sj for each vibrator j=1 to N, a recorded data trace d(t) can be expressed as
                              d          ⁡                      (            t            )                          =                                            ∑                              j                =                1                            N                        ⁢                                                            s                  j                                ⁡                                  (                  t                  )                                            ⊗                                                e                  j                                ⁡                                  (                  t                  )                                                              +                                    n              ⁡                              (                t                )                                      .                                              (        15        )            Using matrices, Equation 15 can be rewritten in the time domain as
                                          d            ⁡                          (              t              )                                =                                                    [                                                                            s                      1                                        ⁡                                          (                      t                      )                                                        ⁢                                                                          ⁢                                                            s                      2                                        ⁡                                          (                      t                      )                                                        ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                                                            s                      N                                        ⁡                                          (                      t                      )                                                                      ]                            ⊗                              [                                                                                                                              e                          1                                                ⁡                                                  (                          t                          )                                                                                                                                                                                                  e                          2                                                ⁡                                                  (                          t                          )                                                                                                                                                ⋮                                                                                                                                                    e                          N                                                ⁡                                                  (                          t                          )                                                                                                                    ]                                      +                          n              ⁡                              (                t                )                                                    ,                            (        16        )            and in the frequency domain as
                                          D            ⁡                          (              f              )                                =                                                    [                                                                            S                      1                                        ⁡                                          (                      f                      )                                                        ⁢                                                                          ⁢                                                            S                      2                                        ⁡                                          (                      f                      )                                                        ⁢                                                                          ⁢                  …                  ⁢                                                                          ⁢                                                            S                      N                                        ⁡                                          (                      f                      )                                                                      ]                            ⁡                              [                                                                                                                              E                          1                                                ⁡                                                  (                          f                          )                                                                                                                                                                                                  E                          2                                                ⁡                                                  (                          f                          )                                                                                                                                                ⋮                                                                                                                                                    E                          N                                                ⁡                                                  (                          f                          )                                                                                                                    ]                                      +                                          N                ⁡                                  (                  f                  )                                            .                                      ⁢                                                      (        17        )            Equation 17 is ill-posed. There is only one record or measurement D(ƒ) from which N outputs, E1(ƒ) to EN(ƒ), need to be separated. Similarly, as long as the number of measurements, i.e. records, is fewer than the number of unknowns, i.e. outputs or seismograms, the separation is ill-posed. Without additional information, it is not possible to devise a filter that can accurately separate the earth responses for each vibrator. Again, the computed M data records obtained by running a modeling program are also described using Equations (15), (16), and (17).
The standard method for obtaining separated seismograms from actual field data is correlation by the specific sweep function for each vibrator. For example, if the field record is correlated with sweep i to isolate the seismogram from location i the result is
                                                                                          g                  i                                ⁡                                  (                  t                  )                                            =                            ⁢                                                                    s                    i                                    ⁡                                      (                                          -                      t                                        )                                                  ⊗                                  d                  ⁡                                      (                    t                    )                                                                                                                          =                            ⁢                                                                    [                                                                                            s                          i                                                ⁡                                                  (                                                      -                            t                                                    )                                                                    ⊗                                                                        s                          i                                                ⁡                                                  (                          t                          )                                                                                      ]                                    ⊗                                                            e                      i                                        ⁡                                          (                      t                      )                                                                      +                                                                                                      ⁢                                                                    ∑                                                                  j                        ≠                        i                                            ,                                              j                        =                        1                                                              N                                    ⁢                                                            [                                                                                                    s                            i                                                    ⁡                                                      (                                                          -                              t                                                        )                                                                          ⊗                                                                              s                            j                                                    ⁡                                                      (                            t                            )                                                                                              ]                                        ⊗                                          e                      j                                                                      +                                                                            s                      i                                        ⁡                                          (                                              -                        t                                            )                                                        ⊗                                                            n                      ⁡                                              (                        t                        )                                                              .                                                                                                          (        18        )            In the frequency domain, Equation 18 can be rewritten as
                                                                                          G                  i                                ⁡                                  (                  f                  )                                            =                            ⁢                                                S                  i                                ⋆                                                      (                    f                    )                                    ⁢                                      D                    ⁡                                          (                      f                      )                                                                                                                                              =                            ⁢                                                                    [                                                                  S                        i                                            ⋆                                                                        (                          f                          )                                                ⁢                                                                              S                            i                                                    ⁡                                                      (                            f                            )                                                                                                                ]                                    ⁢                                                            E                      i                                        ⁡                                          (                      f                      )                                                                      +                                                                                                      ⁢                                                                    ∑                                                                  j                        ≠                        i                                            ,                                              j                        =                        1                                                              N                                    ⁢                                                            [                                                                                                    S                            i                                                    ⁡                                                      (                            f                            )                                                                          ⋆                                                                              S                            j                                                    ⁡                                                      (                            f                            )                                                                                              ]                                        ⁢                                                                  E                        j                                            ⁡                                              (                        f                        )                                                                                            +                                                      S                    i                                    ⋆                                                            (                      f                      )                                        ⁢                                                                  N                        ⁡                                                  (                          t                          )                                                                    .                                                                                                                              (        19        )            The first term in Equations 18 and 19 is the desired seismogram for location i; this can be seen from Equations 3 and 4. The second term is the sum of the cross correlations of the sweeps functions ([Si(ƒ)*Sj(ƒ)]) convolved with the earth responses. The second term will appear as interference noise on the separated record. Note that in this method the extraction of each seismogram is independent of the other seismograms. In contrast, the HFVS method solves for a filter that jointly optimizes the separation for all the seismograms.
In actual field acquisition, the conventional approach to minimize the interference noise shown in Equation 19 is to design sweeps si to SN so they are orthogonal and thus their cross correlation is small. These methods include: using upsweeps and downsweeps (U.S. Pat. No. 4,707,812 to Martinez), using pseudorandom sweeps (see, for example, U.S. Pat. No. 4,168,485 to Payton, et al.), using pseudorandom sweeps with time delays (U.S. Pat. No. 6,704,245 to Becquey), using cascaded sweeps with phase rotations (Moerig et al., International Publication Number WO 02/33442 A2), and using time delays or slip sweeps (see, for example, U.S. Pat. No. 7,050,356 to Jeffryes). Similarly, Ikelle (U.S. Pat. No. 6,327,537) discusses separating or decoding multi-source records for both computer modeling and field acquisition. His methods require either different time delays for each source with the time delays longer than the sweep, or the use of a pseudo-random number (PN) code convolved with each sweep, with the PN codes chosen so that they are mutual orthogonal and that the orthogonality is preserved through the wave propagation. The use of long time delays, longer than a vibroseis sweep, would be equivalent to using multiple sweeps, as defined in the present application, with as many sweeps as there are vibrators. Such long time delays will limit the efficiency gain of using simultaneously operating sources. This method using PN codes, along with the other methods described earlier in this paragraph, rely on cross-correlation (Equation 19) for separating the seismograms.
In practice, however, it is not possible to design sweeps with overlapping frequency bands such that the cross correlations are exactly zero for all the recorded time (that is, they are not perfectly orthogonal, as required by the techniques mentioned in the previous paragraph). Consequently, the separated seismograms obtained from actual field or computer simulated records by using techniques mentioned in the previous paragraph are always corrupted by some amount of interference noise. Even when the sweeps themselves are well-designed, the actual signal imparted into the ground differs from the design sweep in an uncontrolled manner, thereby giving rise to interference noise during separation. To illustrate the interference noise, an earth model with four source locations was convolved with four different pseudorandom sweeps and summed to simulate simultaneous acquisition. The seismograms generated by correlating the data with the four random sweeps are shown in FIGS. 4A-D. The seismograms contain interference noise. As expected, separated seismograms obtained from computer-simulated records would also be corrupted by interference noise.
Various methods have been designed to filter or remove interference noise from separated records obtained from field data. However, all these methods just remove the noise on each individual seismogram and do not move misplaced energy to the appropriate seismogram or optimize the separation. In the slip-sweep method, which uses large time delays between sweeps for different vibrators, interference noise is imprecisely attributed to harmonic noise, which is a type of noise arising from vibrator harmonics. Several methods have been proposed to remove this harmonic noise from separated records using the expected time frequency relationship between the harmonics and fundamental sweep. One approach (Meunier et al., U.S. Pat. No. 6,603,707; Jeffryes, U.S. Pat. No. 6,519,533) uses multiple narrow time-frequency filters to isolate harmonic noise and subtract it from the separated seismograms. Another method (see, for example, Jeffryes, U.S. Patent Publication 2006/0158962 A1) uses correlation of the data with a synthetic sweep containing harmonics and the subtracting the data which correlates with the harmonics. Note that computer-simulated data do not suffer from harmonic noise problems because the sweep signal is exactly known, i.e., the signal driving the vibratory sources is exactly the signal probing the computer model of the earth.
A method called Continuous HFVS (“C-HFVS”) disclosed by Krohn and Johnson (PCT Patent Publication WO 2005/019865) overcomes the requirement to have as many sweeps as there are vibrators, but still uses Equation 10 to derive a filter that optimally recovers the individual seismograms. This method was specifically designed for actual field data. In this method for simultaneous use of multiple seismic vibrators, each vibrator uses continuous segmented sweeps. With C-HFVS, one long data record is recorded as shown in FIG. 5. This illustration can be compared to FIGS. 2A-D. In the case of FIG. 5, there are four vibrators, each sweeping a 32-s segmented sweep. The total data record length is 39 s. To simulate multiple sweeps, the total data are parsed by extracting pieces of the data and vibrator segments. The parsing allows a filter to be designed as in Equation 14. The process is illustrated in FIGS. 6A-D. The vibrator signatures are split into four portions (81, 82, 83, and 84) corresponding to the sweep segments and padded with zeros. The field record is also split into four 16-second sections (85, 86, 87, 88) corresponding to the four sections from FIG. 5 (65, 66, 67, 68). It may be noted that pieces of the data are reused. The data segment 91 is the same as segment 86, segment 92 is the same as segment 87, and segment 93 is the same as 88. Compared to HFVS data with four separate sweeps followed by a listening time in which the vibrator motion ceases (FIG. 2, 50), the times 91, 92, and 93 in FIGS. 6A-D are contaminated with the strong events from the next segment. In addition, the start of each segment will contain some data from the previous segment because the deeper reflection's response to the earlier sweep segment will arrive at the same time as the shallow reflection's response to the later sweep segment. After separation (FIGS. 7A-D), interference noise can be observed at the end of the segment length (200) that bleeds upward.
There continues to be a need for a method that satisfactorily separates the responses for two or more geophysical sources operating simultaneously with minimal separation noise corrupting the separated responses when the number of sweeps used are fewer than the number of sources. The present invention addresses this need.